Some time ago, I wrote this in response to a StackOverflow question, but wanted to share here on the blog.

The question basically asked how you could make a floating point operation the same between the CPU and the GPU, and here is an updated version of the answer:

There are many reasons why it is not realistic to expect the same results from floating point computations run on the CPU and GPU. It’s much stronger than that: you can’t assume that FP results will be the same when the same source code is compiled against a different target architecture (e.g. x86 or x64) or with different optimization levels, either.

In fact, if your code is multithreaded and the FP operations are performed in different orders from one run to the next, then the EXACT SAME EXECUTABLE running on the EXACT SAME SYSTEM may produce slightly different results from one run to the next.

Some of the reasons include, but are not limited to:

floating point operations are not associative, so seemingly-benign reorderings (such as the race conditions from multithreading mentioned above) can change results;

different architectures support different levels of precision and rounding under different conditions (i.e. compiler flags, control word versus per instruction);

different compilers interpret the language standards differently, and

some architectures support FMAD (fused multiply-add) and some do not.

Note that for purposes of this discussion, the JIT compilers for CUDA (the magic that enables PTX code to be future-proof to GPU architectures that are not yet available) certainly should be expected to perturb FP results.

You have to write FP code that is robust despite the foregoing.

As I write this today, I believe that CUDA GPUs have a much better-designed architecture for floating point arithmetic than any contemporary CPU. GPUs include native IEEE standard (c. 2008) support for 16-bit floats and FMAD, have full-speed support for denormals, and enable rounding control on a per-instruction basis rather than control words whose settings have side effects on all FP instructions and are expensive to change.

In contrast, CPUs have an excess of per-thread state and poor performance except when using SIMD instructions, which mainstream compilers are terrible at exploiting for performance (since vectorizing scalar C code to take advantage of such instruction sets is much more difficult than building a compiler for a pseudo-scalar architecture such as CUDA). And if the wikipedia History page is to be believed, Intel and AMD appear to have completely botched the addition of FMAD support in a way that defies description.

You can find an excellent discussion of floating point precision and IEEE support in NVIDIA GPUs here.